#6353072

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353061
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3014 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

(2,7)(3,11)(4,15)(8,21)(9,19)(10,16)(12,14)(17,29)(20,24)(22,31)(23,32)(25,30);(1,2,5,10)(3,8,11,24)(4,16,15,7)(6,9,18,25)(12,22,14,23)(13,21,26,20)(17,30,29,19)(27,31,28,32);(4,17)(7,19)(13,28)(15,29)(16,30)(20,31)(21,32)(26,27);(1,3,5,11)(2,8,10,24)(4,13,15,26)(6,14,18,12)(7,20,16,21)(9,23,25,22)(17,28,29,27)(19,31,30,32);(3,12)(8,22)(11,14)(13,27)(20,32)(21,31)(23,24)(26,28)

Polycyclic presentation:

2^9|b^2=h,d^2=h,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260268);